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ArticleOriginal scientific text

Title

Optimal edge ranking of complete bipartite graphs in polynomial time

Authors 1

Affiliations

  1. Department of Information Management, Nan-Kai Institute of Technology, Tsao-Tun, Nantou 542, Taiwan

Abstract

An edge ranking of a graph is a labeling of edges using positive integers such that all paths connecting two edges with the same label visit an intermediate edge with a higher label. An edge ranking of a graph is optimal if the number of labels used is minimum among all edge rankings. As the problem of finding optimal edge rankings for general graphs is NP-hard [12], it is interesting to concentrate on special classes of graphs and find optimal edge rankings for them efficiently. Apart from trees and complete graphs, little has been known about special classes of graphs for which the problem can be solved in polynomial time. In this paper, we present a polynomial-time algorithm to find an optimal edge ranking for a complete bipartite graph by using the dynamic programming strategy.

Keywords

ENG
graph algorithms, edge ranking, vertex ranking, edge-separator tree, complete bipartite graphs

Bibliography

  1. B. Aspvall and P. Heggernes, Finding minimum height elimination trees for interval graphs in polynomial time, BIT 34 (1994) 484-509, doi: 10.1007/BF01934264.
  2. H.L. Bodlaender, J.S. Deogun, K. Jansen, T. Kloks, D. Kratsch, H. Müller and Z. Tuza, Rankings of graphs, SIAM J. Discrete Math. 11 (1998) 168-181, doi: 10.1137/S0895480195282550.
  3. C.C. Chou, C.M. Fu and W.C. Huang, Decomposition of Km,n into short cycles, Discrete Math. 197/198 (1999) 195-203.
  4. D.G. Corneil, H. Lerchs and L.K. Stewart, Complement reducible graphs, Discrete Appl. Math. 3 (1981) 163-174, doi: 10.1016/0166-218X(81)90013-5.
  5. P. De La Torre, R. Greenlaw and A.A. Schaffer, Optimal edge ranking of trees in polynomial time, Algorithmica 13 (1995) 592-618, doi: 10.1007/BF01189071.
  6. J.S. Deogun, T. Kloks, D. Kratsch and H. Müller, On vertex ranking for permutation and other graphs, Lecture Notes in Comput. Sci. 775 (Springer-Verlag, 1994) 747-758.
  7. J.S. Deogun, T. Kloks, D. Kratsch and H. Müller, On vertex ranking for trapezoid, circular-arc and other graphs, Discrete Appl. Math. 98 (1999) 39-63, doi: 10.1016/S0166-218X(99)00179-1.
  8. P.E. Haxell, Partitioning complete bipartite graphs by monochromatic cycles, J. Combin. Theory (B) 69 (1997) 210-218, doi: 10.1006/jctb.1997.1737.
  9. A.A. Ivanov, R.A. Liebler, T. Penttila and C.E. Praeger, Antipodal distance-transitive covers of complete bipartite graphs, Europ. J. Combinatorics 18 (1997) 11-33, doi: 10.1006/eujc.1993.0086.
  10. A.V. Iyer, H.D. Ratliff and G. Vijayan, On an edge ranking problem of trees and graphs, Discrete Appl. Math. 30 (1991) 43-52, doi: 10.1016/0166-218X(91)90012-L.
  11. T. Kloks, H. Müller and C.K. Wong, Vertex ranking of asteroidal triple-free graphs, Inform. Process. Lett. 68 (1989) 201-206, doi: 10.1016/S0020-0190(98)00162-8.
  12. T.W. Lam and F.L. Yue, Edge ranking of graphs is hard, Discrete Appl. Math. 85 (1998) 71-86, doi: 10.1016/S0166-218X(98)00029-8.
  13. T.W. Lam and F.L. Yue, Optimal edge ranking of trees in linear time, Algorithmica 30 (2001) 12-33, doi: 10.1007/s004530010076.
  14. C.E. Leiserson, Area efficient graph layouts for VLSI, in: Proceedings of the 21st Annual IEEE Symposium on Foundations of Computer Science FOCS'80 (1980) 270-281.
  15. C.M. Liu and M.S. Yu, An optimal parallel algorithm for node ranking of cographs, Discrete Appl. Math. 87 (1998) 187-201, doi: 10.1016/S0166-218X(98)00056-0.
  16. D.C. Llewellyn, C. Tovey and M. Trick, Local optimization on graphs, Discrete Appl. Math. 23 (1989) 157-178, doi: 10.1016/0166-218X(89)90025-5.
  17. A. Pothen, The complexity of optimal elimination trees, Technical Report CS-88-13 (the Pennsylvania State University, 1988).
  18. A.A. Schäffer, Optimal node ranking of trees in linear time, Inform. Process. Lett. 33 (1989/1990) 91-96.
  19. P. Scheffler, Node ranking and searching on graphs (Abstract), in: U. Faigle and C. Hoede (eds.), 3rd Twente Workshop on Graphs and Combinatorial Optimization, Memorandum No. 1132, Faculty of Applied Mathematics, University of Twente, The Netherlands, 1993.
  20. J.D. Ullman, Computational aspects of VLSI (Computer Science Press, Rockville, MD, 1984).
  21. X. Zhou and T. Nishizeki, Finding optimal edge rankings of trees, in: Proceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms SODA'95 (1995) 122-131.
Discussiones Mathematicae Graph Theory
2006/26/1
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Pages:
149-159
Main language of publication
English
Received
2005-06-17
Accepted
2005-10-20
Published
2006

DOI: 10.7151/dmgt.1309

Exact and natural sciences